Lecture 4Introduction to Boolean Algebra

Binary Operators

In the following descriptions, we will let A and B be Boolean variables and define a set of binary operators on them. The term binary in this case does not refer to base-two arithmetic but rather to the fact that the operators act on two operands.

unary operator NOT

binary operators AND, OR, NAND, XOR

Logic Gates

NOT

AND

OR

XOR

NAND

NOR

F(A,B,C) = A + BC'

Truth Tables

A B C C' BC' A+BC'

0 0 0 1 0 0

0 0 1 0 0 0

0 1 0 1 1 1

0 1 1 0 0 0

1 0 0 1 0 1

1 0 1 0 0 1

1 1 0 1 1 1

1 1 1 0 0 1

A Boolean Function Implemented in a Digital Logic Circuit

Pow

er S

uppl

y

Input = 1 1 0

Voltmeter

NOT gates AND gates

OR gate(s)

The Part of the Circuit Usually Not Shown

A One-Bit Adder Circuit

Venn Diagrams

A A B A B

A B A BA B

A

~A + B

A .BA+B

A .BA=B

A A B A B

A B A BA B

A

~A + B

A .BA+B

A .BA .BA=B

Three-Variable Venn Diagram

F(A,B,C) = A + BC'

000

001

010

011

100

101

110

111

A B

C

A B

C

De Morgan's Theorem

BABA

BABA

A B A+B ~(A+B) ~A ~B (~A).(~B) ~(A+B)=(~A).(~B)

0 0 0 1 1 1 1 1

0 1 1 0 1 0 0 1

1 0 1 0 0 1 0 1

1 1 1 0 0 0 0 1

A B A+B ~(A+B) ~A ~B (~A).(~B) ~(A+B)=(~A).(~B)

0 0 0 1 1 1 1 1

0 1 1 0 1 0 0 1

1 0 1 0 0 1 0 1

1 1 1 0 0 0 0 1

Textbook Reading for Chapter 4